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SOVIET PHYSICS JETP
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VOLUME 29, NUMBER 4
OCTOBER, 1969
MEASUREMENT OF THE ELECTRON MEAN FREE PATH IN POTASSIUM BY MEANS OF
THE RADIO FREQUENCY SIZE EFFECT
V. S. TSOI and V. F. GANTMAKHER
Institute of Solid State Physics, USSR Academy of Sciences
Submitted November 20, 1969
Zh. Eksp. Teor. Fiz. 56, 1232-1241 (April, 1969)
The sensitivity of the method with respect to changes in the mean free path l can be increased by increasing the sample thickness d. The behavior of the size effect line for l < d requires an additional
investigation. The change in the line shape due to an almost tenfold change in the sample thickness
showed that for a relatively small mean free path d the relation between the line shape and field distribution in the skin layer is different from that for l >> d. Some specific conditions (formation of
lines by electrons moving along trajectories of various length) are found, which lead to a particularly
strong dependence of the line shape on temperature. The temperature-dependent part of the reciprocal path, which defines the line amplitudes, is measured. It is proportional to T 3 • In this connection
the influence of the collision site on the electron trajectory and the direction of the phonon momentum
on the efficiency of electron-phonon collisions is discussed. The residual resistance, whose temperature-dependent part is proportional to T 6 , was measured for the same samples by the rotating magnetic field technique.
INTRODUCTION
INTEREST in the measurements of the anisotropy of
the mean free path l of the electrons on the Fermi
surface has stimulated the development of new methods
for measuring l on the basis of effects heretofore used
mainly for the study of the energy spectrum. With respect to the radio-frequency size effect (SE), this
means a changeover from the study of the locations of
the SE lines to a measurement of their amplitudes.
Such measurements were already made for tin(l, 2 J,
indium [31, gallium [4 ], cadmium [sl, and bismuth (sJ. In
this paper we investigate certain questions connected
with the developement of a method for measuring l of
potassium on the basis of the SE. Potassium was
chosen because its Fermi surface is, within 0.1%, a
sphere of radius PF =0.813 x 10-19 g-cm/sec(7l, and
the SE on potassium was itself investigated in sufficient
detail[ 8 ' 91 •
The most convenient for the measurement of l are
those SE which admits of only a single passage of the
electrons through the "receiving" skin layer, for example the SE at the limiting point. The amplitude A of
the lines of such SE is proportional to the number of
electrons traversing without scattering the path :\ from
one side of the plate to the other:
(1)
( d is the thickness of the sample, and the proportionality coefficient a can be readily determined from
geometrical considerations in each concrete case-see
below). As seen from (1 ), to increase the sensitivity of
A to changes of l it is necessary to work in the region
l<f..,
(2)
although this leads to a very strong decrease of the
absolute value of A. The condition (2) makes it possible to apply formula (1) also to the SE on the central
663
cross section. Although in principle a multiple passage
of the electron through the skin layer is possible in this
SE, and the amplitude should be determined by an infinite sum of terms of the type (1 ), nevertheless, owing
to the smallness of the free path, only half of the trajectory loops make a contribution to the effect (in the
case of a spherical Fermi surface a = 1r/2 ).
In all the prior investigations of the SE line shape,
the tendency was to satisfy the inverse condition l >> :\.
We have therefore undertaken certain additional measurements of the line shape under conditions (2 ). The
results constitute the content of the first part of this
paper.
The different mechanisms for electron scattering
can be assumed to be independent in the first approximation, and l can be represented in the form
1/ l
=
1/ lo+ 1/lr(T),
(3)
where lo is the mean free path at OoK and is caused
by the static violations of the lattice periodicity, while
lT is the free path determined by the electron-phonon
collisions. As is well known, in such collisions the
electron is scattered through a small angle {:3 ~ PphiPF
( Pph is the phonon momentum) .. On the other hand, each
type of SE has its own angle 8< 1 > of "maximum possible scattering,'' and when this angle is exceeded the
electron ceases to contribute to the amplitude of SE
line. Inasmuch as f3 is comparable with e<i>, the efficiency of the electron-phonon collisions may turn out
to be different for different types of SE. Then the
measured numerical values of lT will also be different.
Moreover, even for a concrete type of SE, the value of
e<i> depends on the ratio o/ d ( o-depth of skin layer),
and sometimes also on the magnitude of the magnetic
field H[ 2 J, which may affect the measurements of l/T.
Since interest attaches not to the functions lt ( T) in
each concrete case, but to the path le, ph(T) traversed
by the electron between two collisions with phonons, we
664
V. S. TSOI and V. F. GANTMAKHER
have attempted to ascertain under what conditions l T
can be reliably identified with l e, ph· For a comparison
of the numerical results, we measured the statistical
electric resistivity p ( T) on the same samples as used
for the observation of the SE, using the method of rotating magnetic fieldsr•o,uJ.
EXPERIMENTS
Although the SE manifests itself in the same manner in the real and imaginary parts of the surface impedance of plates Z =R + iX, it is more convenient to
use for amplitude measurements the quantity X. The
sensitivity of apparatus to the measurement of R is
determined by the amplitude generated by the radial
oscillator in whose tank circuit the investigated sample is connected, and requires a constant correction
when the temperature or the magnetic field is varied.
To the contrary, the changes of the generation frequency D.f ~ -b.X do not depend on the generation regime, and no calibration is required for relative measurements of the amplitude of the SE lines.
The apparatus employed is described inr 121 • Modulation coils were mounted independently of the electromagnet, so that the modulating field did not rotate
relative to the surface of the sample when the inclination of the magnetic field was varied. Since the smallest tilt of the coils led to the appearance of a signal
af/ aH that increased monotonically with increasing
field H, the modulating field was positioned parallel
to the surface of the sample to within a fraction of a
degree by minimizing the signal in the strong field.
When desired, the tilt of the coils could be used to
compensate for the monotonic part of the change of the
impedance in the vicinity of the SE line.
The potassium was obtained from KN0 3 reduced
with iron with subsequent vacuum distillation. The
distillation was in a glass sphere of approximately
12 mm diameter, which was sealed off and placed on a
torsion balance with a rotating magnetic field to measure the resistancer 10 • 111 • The field ranged from 5 to
15 Oe, the rotary frequency at room temperature was
of the order of 100 Hz, and at helium temperatures on
the order of 0.02-0.002 Hz. The measurement accuracy
at low temperatures was 0.5%.
The samples were made either by pouring liquid
metal in a dismountable glass mold (samples 1 and 3)
or by compressing; a piece of potassium between two
glass plates. In either case, the glass was coated with
a layer of mineral oil to prevent the metal from sticking to the glass when solidified. The prepared samples
were washed in gasoline, and then the gasoline was
evacuated, and the sample was cooled in vacuum to the
temperature of liquid nitrogen, at which it was customarily stored. In the instrument, the samples were
inserted in the apparatus likewise at liquid-nitrogen
temperature. After the high-frequency measurements
were completed, samples 1 and 3, which had the form
of regular discs of 18 mm diameter, were again
mounted on a torsion balance. This made it possible
not only to verify that the potassium was not contaminated during the course of preparation of the samples,
but also to measure directly the value of p ( T) of the
sample with which the SE was investigated. The char-
acteristics of the samples are listed in the table. For
the purest samples Proom!Po ~ 6 x 10 3 (po-residual
resistance).
Line Shape
The results of the study of the line shape can be
summarized as follows. At relatively small free path
(l < d, ,\), the SE line shape can vary with the ratio
between the parameters 6, l, and d.
The dependence on l can be demonstrated most
easily in Fig. 1 for the SE line in a field H inclined to
the surface of the metal at an angle cp = 25°. As shown
in [9 1 and [l 4 l, at an inclination angle cp ~ 2 5°, the SE
line from a spherical Fermi surface is formed as a
result of the focusing of a large group of electrons
moving along different trajectories and arriving from
one side of the plate to the other along paths of different lengths, so that b.,\/,\~ 0.4 (see Fig. 2; ,\max
= 2d/ sin 2 cp ). At 4 °K, the SE line is formed mainly as
a result of the shorter trajectories of type b. Lowering
of the temperature causes the electrons to start to arrive at the opposite side of the plate also along longer
trajectories a, as a result of which the line shape
changes.
iJf/tl H
FIG. I. Plot of SE lines for sample
I, <p = 25°. Modulation field parallel to
the surface of the metal. Projection of
H on the plane of the sample is parallel to the high-frequency current j. Frequency f = 4.1 MHz. The lower curve
is plotted with the sensitivity increased
by a factor of 2.
FIG. 2. Curve a- boundary orbit separating the effective orbits from
the ineffective ones [ 14 ) , b-orbit with two effective points per period.
On the Fermi sphere, this is the only case of focusing of trajectories of different length, although for an
arbitrary Fermi surface such a focusing should probably be encountered quite frequently. All the remaining SE lines observed by us for potassium did not
MEASUREMENT OF THE ELECTRON MEAN FREE PATH IN POTASSIUM
Sample numberjd, mm•l
z~P), mm
'
1
2
3
4
5
6
FIG. 3. Plots of SE lines of
samples I, 4, and 5, tp = 0, j 1 H,
f""' 7 MHz, T = 1.3° K. The ordinate scales are different.
0.2
{/.§
1,{1
0.63
0,23
0,24
0.26
0,09
0~2:j
II,,
0.19
0.1:l
0.13
0.13
0,13
0,06
mm
0.12
0,16
I
665
10' &/1
10' &fd
3.2-2.6
6,5
4.1
3,8
3,6
8.5
1.0-0.8
3. 7
2.2
1,\J
5,2
2,0
Note. The depth of penetration 8 was determined from theSe line width: 8"'
"'0.15 dliH/H [3 •13 ].
/. 4 /1, kOe
change their shape with temperature. However, in
measurements of l ( T) this circumstance must always
be checked, for other mechanisms governing the dependence of the line shape on T are quite probable.
For example, for sufficiently broad lines t.H/Ho ~ 10%
(H 0 -field corresponding to the left edge of the line), an
important role may be played by the fact that in the
right part of the line, in a field H =Ho + AH, the effective trajectory connects one of the surfaces of the plate
not to the other surface, but to the internal part of the
skin layer located near it, so that the role of d is
played by the quantity d - o ~ d( 1 - t.H/Ho ). Because
of this, the right-hand and left-hand extrema of the line
could, generally speaking, vary with the temperature in
different manners. In our experiments we did not observe such an effect, probably because the broad lines
were obtained by us at not too small values of d, when
condition (2) was violated.
There exists also other causes for changes of the
line shape. As seen from Fig. 3, the SE line shape is
different for samples of different thickness. These
changes are not connected with the multiplicity of the
return of the electron to the skin layer and are inherent
apparently in SE of all types. This indicates, in particular, that for all the samples inclination of the field by
~15 o did not change the shapes of the lines shown in
the figure, and only shifted them towards stronger
fields [81 • It should be particularly noted that in two
samples, 4 and 6, the purity of which differs by a factor of 2, the SE lines obtained for equal o and d were
identical in shape, although their amplitudes differed
by one order of magnitude.
Of course, the difference in l between samples 4
and 6 is not so appreciable as to influence noticeably
the distribution of the electromagnetic field in the skin
layer. On the other hand, however, it is not likely that
this distribution can be influenced also by the interaction between the skin layers under conditions when
o/d"" (1-2)x10- 2 (seethetable). Itfollowstherefore that when l < d the SE line shape does not reflect
the distribution of the field in the skin layer, as is the
case in the opposite limiting case l >> d (compare
with[ 13 l ), and depends on some complicated relations
between the parameters o, d, and l. The meaning of
this conclusion is seen particularly clearly from Fig.
4, where plots of two SE lines in samples of different
FIG. 4. Comparison of Se lines
of samples I and 4, <P = 0, j 1 H,
f""' 6 MHz, T = 1.9° K. The ordinate scales are different.
thickness (samples 1 and 4) are drawn in abscissa
scales proportional to d- 2 (we recall that the line
width is t.H ~ H0 o/d ~ o/d 2 ).
Measurements of Free Path
a) Results. The measurements of l 0 , obtained by
determining the dependence of the amplitude of SE
lines at the limiting point on the angle cp of inclination
of the magnetic field to the surface of the sample, were
performed on sample 4 in the angle interval 14.5 o < cp
< 20° and on sample 5 in interval8.5o < cp < 14.5°.
The measurements were made at T = 1.3 °K, when the
phonon part of the scattering is no longer significant.
Reduction by means of the formula
A= Aoexp (-'J../lo) = Ao exp (-2d I lo sin 2cp)
led to values of lo that coincided practically with the
values of zo(p) obtained from measurements of the
static conductivity (see the table; Zo(P) was calculated
from the residual resistance p 0 by means of the
formula l (p) = PFie~po, and Zo was obtained from
SE experiments). This agreement means that the SE
and the de conductivity are equally sensitive to scattering by static inhomogeneities.
Typical results of the measurement of lT by the
SE method are shown in Fig. 5 in coordinates
2( lnA)/7Td = const- 1/lT and T 3 • It is seen from the
figure that all the samples except the thinnest sample
5 lead to a relation ZT =bT-\ whereas for the first SE
line on the central section (trajectory diameter
D =d) there were obtained on samples 1-4 values of
b from 2.7 to 3.0 cm-deg 3 •
The lower two straight lines of Fig. 5 pertain to SE
lines on the central section in double and triple fields
666
V. S. TSOI and V. F. GANTMAKHER
PfProorrfD'
z.q
FIG. 5. Temperature dependence
of the amplitude of the SE lines, measured in a parallel field. The zero on
the ordinate axis pertains to the
straight line drawn through all the
symbols 0. The random measurement error can be assessed from the
scatter of the points.
2
6
8
10
IZ
J
T 5 ·10· 2 txJ
/Q
5
6
r 6 ·tu-1!+J
FIG. 6. Electricresistivityp as the function of T 5 (X) and T6 (+).
The p(T 5 ) curve is shifted upward along the ordinate axis by a distance S.
ZU
QU
50
TJ
6
(D =d/2 and d/3). When reduced by means of formula
(1) with a = rr/2, they yielded for b the values 1.5 and
1.0 cm-deg 3 , i.e., smaller by factors 2 and 3 respectively than the values measured on the first line. From
the temperature dependence of the amplitude of the SE
line on a section close to the central one, at an inclination angle cp = 17 o, the value obtained for sample 4 was
b = 1.7 cm-deg 3 •
In sample 5 the relation is in the form lT ~ T-4 , not
only in the central section but also at the limiting point.
The results of the measurements of the resistance
p ( T) of a spherical sample of potassium sealed in glass
are shown in Fig. 6. Since the shape of the sample
necessary to obtain the absolute values of the resistance by this method must be regular, we confine ourselves to relative measurements. The absolute values
of p were obtained by starting from the value Proom
= 7.19 x 10-6 n-cm (see[lsJ ). As seen from Fig. 6, in
the interval 1.3-4 . 2 oK the resistivity of potassium
varies like p ( T) ~ T 6 • The T 6 dependence was observed already in the measurements of p ( T) of pure
sodium [16 l. The faet that the T 5 dependence has been
observed so far for potassium [1 6 • 17 l is apparently due
to the insufficient purity of the investigated material.
b) Discussion. The cubic dependence in lT( T) is
usually evidence that a single collision with a phonon
makes an electron ineffective. At the same time, the
customarily employed estimate for the angle of the
"maximum permissible scattering" e(c) on the
central section, namely e(c) ~ IF7d, leads to the inequality e(c) > {3, which should result in the relation
lT ~ T- 5 , owing to the diffuse character of the scattering.
Let us examine, however, in greater detail the different possibilities of scattering of an electron on the
central section by a phonon. Figures 7a and b show
three essentially di.fferent cases: the electron goes
over to the neighboring orbit in momentum space without changing the phase of rotation {process 1 ), it shifts
along the previous orbit remaining in the vicinity of the
point Vz = 0 (process 2; the z axis is normal to the
surface of the sample) or in the vicinity of the point
Vz = Vz.max {process 3). The effectiveness of the process 1 is determined by the change of the diameter of
the trajectory D = d - o = d( 1 - lif/2 ), whence 11 1
~ ..f6Td. The process 2, the effectiveness of which is
6~-6 --
--
11•
e
If
/IO
FIG. 7. Drawings a and d pertain to momentum space, while b, c,
and e pertain to coordinate space. The dashed Jines designate the skin
layers and the trajectories of the electrons after scattering.
connected with the change of the time of stay in the skin
layer, i.e., with the width of the effectiveness strip
(Fig. 7b), leads to a similar estimate 11 2 ~ 1!1. On the
other hand, the effectiveness of process 3 is determined by the shift of the center of the trajectory along
the direction of the electron velocity at the instant of
scattering, and it can be readily seen from Fig. 7b that
e3 ~ o/d.
It follows from the foregoing that in the vicinity of
the point Vz = Vz.max ~ I v I there exists on the central
trajectory a region of most effective scattering. In our
experiments we have the relation
o/d~r""/p.-<ybl"d.
(4)
Therefore the observed lT ~ T- dependence is due
apparently only to scattering processes of the third
type, whereas the remaining collisions with the phonons
are not very effective. For the thinnest sample (sample 5 ), the left side of the inequality (4) is violated and
the scattering acquires a diffuse character already on
the entire trajectory, and consequently the degree of T
increases.
It is curious that Naberezhnykh and Tsymbal ( 51 , who
obtained lT ~ T- 5 for cadmium, dealt with a "lenslike"
trajectory with a kink, in which Vz « I vI throughout
(Fig. 7c), and consequently scattering processes of the
third type are not very effective.
All the foregoing is applicable also to the SE at the
I turning point. The estimate e<OJ ~ (on/d)2/ 3 cp (n = 1,
3
MEASUREMENT OF THE ELECTRON MEAN FREE PATH IN POTASSIUM
2, 3, ... is the number of the line), given in l2l, starts
from the width of the effectiveness band and is valid
only for processes 1 and 2 (Fig. 7b), whereas for
processes 3 (Fig. 7e ), which shift the axis of the trajectory, we have e~o> ~ (6n/d)rp. The difference here,
however, is somewhat smaller than on the central section, and furthermore all the estimates of e<o> contain
an additional small factor rp, which made it possible to
obtain in [1 • 2 l effectiveness of all the collisions.
It might seem that in the approximation (2) the amplitudes of the SE lines on the central section in multiple fields should depend on l in the same manner as
the first line. Indeed, when D =d/n (n = 1, 2, 3, ... )
the field amplitude in the burst closest to the surface is
q ~ exp( -rrd/2n), and the amplitude of the SE line is
A~ qn r18 l. However, for sample 1, in the temperature
interval 3.9-1.3°K, the first SE line in a parallel field
increased by six times, the second by 2.6 times, and
the third by only 1.9 times. The same values of lT
for different n are obtained by reducing the experimental data in accordance with formula (1) with a = rr/2n,
but in spite of the attractiveness of such a reduction
method, there seems to be no justification for it. Inasmuch as for the first number at different values of d
the results agree with each other, the discrepancy for
different values of n cannot be ascribed to the change
of the parameter 6/D with changing number. It is possible that the measurements made at n "" 1 cannot be
reduced in principle by means of formula (1 ), since
the probability that the electron will execute one revolution without being scattered when D = d is equal to
the probability of executing n revolutions at D = d/ n,
and the amplitudes of SE lines of all numbers are of
the same order.
In view of the lack of a reliable method of reducing
the amplitudes of the lines with higher numbers, we
confine ourselves to a consideration of the values of b
obtained with lines with n = 1, namely b = 2.8 cm-deg 3 •
Inasmuch as we obtained a T 6 dependence for the electric resistance, which is unexpected from the theoretical point of view, let us compare first the results obtained for b with the estimates of the electron-phonon
range l (K), obtained from data on the electronic heat
conductivity K( 1 B] and the specific heat c[20]: Ze,ph
~ l (K) ~ 3K/ Cv ~ 0.35 T- 3 em (the Fermi velocity is
v = 0.74 x 10 8 em/sec). Starting from the fact that the
value of lT observed by us in the SE is due to collisions of type 3, amounting to approximately Y4 of the
total number of collisions, we obtain le,ph r::; lTI 4
r::; 0. T 3 em. The discrepancy between our results in
parallel and inclined fields' is apparently due to the
change of the fraction of the effective collisions. Taking into account the approximate character of all the
numerical estimates, the agreement between our results and z(K) should be regarded as perfectly satisfactory. Thus, we have succeeded in measuring in
potassium the average free path of the electron between two collisions with phonons with the aid of the
SE on the central cross section. We hope that in the
future the availability of purer potassium will enable
us to verify once more the results with the aid of the
SE at the limiting point.
We now turn to the temperature dependence of the
static resistance. The only mechanism to us, leading
667
to a stronger temperature dependence of the electric
resistance than T 5 , is connected with the quenching of
the U-processes of scattering in the electron-phonon
system [21 l. It is not at all surprising that at temperatures lower than 40°K the U-processes have not yet
been fully quenched out, since, in view of the fact that
the minimum distance from the Fermi Sphere to the
boundary of the Brillouin zone is P = 0.173 tirr/a (alattice constant), and the velocity of the slow transverse wave along the corresponding direction (the
[110] axis) is s = 0.65 x 10 5 cm/sec[ 22 l, we need a
phonon with energy 2Ps ~ 10° for the Unklapp process.
However, if we turn to the numerical values of the free
path, additional difficulties arise. If the phonon system
is in equilibrium, the quantity
entering in the
formula
df)
should be equal to
z!fl ::::; le,ph(8/T) 2 ::::; 4·10 3 T-5 cm
( ® r::; 90° is the De bye temperature). If the p ~ T 6 dependence is due to the quenching of the U-processes,
this means that there is dragging by the phonon system.
But in this case the effectiveness of the electron-phonon collisions should decrease, so that z(f) should become still larger at all temperatures. OUr experimental result is of the form
z':f> = 3.4 ·10'T-' em.
df)
so that even at 1oK the value of
is one-tenth the
value of le ph(e/T) 2 , instead of being larger. Similar
discrepancies between z(K) and df) in alkali metals
were already pointed out by Klemens [23 1. Discrepancies of the same sign were observed also for other
metals [2 ' 3 l.
The authors are grateful to Academician P. L.
Kapitza for the opportunity of working at the Institute
of Physics Problems, to Professor Yu. V. Sharvin for
help in organizing the work and for valuable advice and
discussions, to G. M. Kolosova for purifying the KN0 3
salt, and to I. F. Poletaev for consultations on the
procedure of obtaining potassium.
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V. S. TSOI and V. F. GANTMAKHER
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Translated by J. G. Adashko
143